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We establish the asymptotic behaviour of the least energy solutions of thefollowing nonlocal Neumann problem:  \begin{align*} \left\{\begin{array}{l l} { d(-\Delta)^{s}u+ u= \abs{u}^{p-1}u} \text{ in $\Omega,$ }  { \mathcal{N}_{s}u=0 } \text{ in $\mathbb{R}^{n}\setminus\overline{\Omega},$}  {u>0} \text{ in $\Omega,$} \end{array} \right.\end{align*} where $\Omega\subset \mathbb{R}^{n}$ is a bounded domain of class $C^{1,1}$,$1<p<\frac{n+s}{n-s},\,n> \max \left\{1, 2s \right\}, 0<s<1,\,d>0$ and$\mathcal{N}_{s}u$ is the nonlocal Neumann derivative. We show that for small$d,$ the least energy solutions $u_d$ of the above problem achieves$L^{\infty}$ bound independent of $d.$ Using this together with suitable$L^{r}$-estimates on $u_d,$ we show that least energy solution $u_d$ achievemaximum on the boundary of $\Omega$ for $d$ sufficiently small.

Asymptotic behaviour of the least energy solutions of fractional semilinear Neumann problem